Find all values of x that make the function f(a) a local minimum and (b) a local maximum. f^'(x)

Find all values of x that make the function f(a) a local...

Key Concepts

Critical Points

In differential calculus, critical points are those points in the domain of a function where its first derivative is either zero or undefined. They are the primary candidates for hosting local extrema such as minima or maxima. However, having a critical point does not guarantee the presence of an extremum; further analysis is required to classify the point.

First Derivative Test

The first derivative test is used to classify critical points by examining the sign changes of the derivative as one passes through these points. If the derivative changes from negative to positive at a critical point, the function has a local minimum there; if it changes from positive to negative, the function has a local maximum. When there is no sign change, the test indicates neither a local minimum nor a local maximum.

Monotonicity

Monotonicity refers to the property of a function being exclusively non-decreasing or non-increasing over an interval. When the derivative of a function is non-negative (or non-positive) over an interval—with strict positivity except possibly at isolated points—the function is monotonic. In such cases, isolated zeros in the derivative (critical points) often do not correspond to local extrema because the function does not change its overall increasing or decreasing behavior around those points.

Second Derivative Test

The second derivative test is a method for further investigating the nature of a critical point by analyzing the concavity of the function at that point. If the second derivative is positive, the function is concave upward and the critical point is typically a local minimum; if it is negative, the function is concave downward and the critical point is typically a local maximum. However, when the second derivative is zero at the critical point, the test is inconclusive and higher-order derivatives or additional analysis must be used.